3.44 \(\int \frac{\sqrt{c+d x} \sqrt{e+f x} (A+B x+C x^2)}{a+b x} \, dx\)

Optimal. Leaf size=450 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)+16 a^3 C d^3 f^3-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )+b^3 \left (-\left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right )}{8 b^4 d^{5/2} f^{5/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{8 b^3 d^2 f^2}-\frac{2 \sqrt{b c-a d} \sqrt{b e-a f} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^4}-\frac{\sqrt{c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f} \]

[Out]

((4*b*d*f*(2*A*b*d*f - a*C*(d*e + c*f)) + (b*d*e - b*c*f + 4*a*d*f)*(2*a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f)))
*Sqrt[c + d*x]*Sqrt[e + f*x])/(8*b^3*d^2*f^2) - ((2*a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f))*Sqrt[c + d*x]*(e +
f*x)^(3/2))/(4*b^2*d*f^2) + (C*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*d*f) - ((16*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f
^2*(C*d*e + c*C*f + 2*B*d*f) - 2*a*b^2*d*f*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)) - b^3*(C*(d*e -
c*f)^2*(d*e + c*f) - 2*d*f*(B*(d*e - c*f)^2 - 4*A*d*f*(d*e + c*f))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]
*Sqrt[e + f*x])])/(8*b^4*d^(5/2)*f^(5/2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*Sqrt[b*e - a*f]*ArcTanh
[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/b^4

________________________________________________________________________________________

Rubi [A]  time = 1.36945, antiderivative size = 453, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1615, 154, 157, 63, 217, 206, 93, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)+16 a^3 C d^3 f^3-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )+b^3 \left (-\left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right )}{8 b^4 d^{5/2} f^{5/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left (\frac{(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e))}{b d f}-4 a C (c f+d e)+8 A b d f\right )}{8 b^2 d f}-\frac{2 \sqrt{b c-a d} \sqrt{b e-a f} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^4}-\frac{\sqrt{c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x),x]

[Out]

((8*A*b*d*f - 4*a*C*(d*e + c*f) + ((b*d*e - b*c*f + 4*a*d*f)*(2*a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f)))/(b*d*f
))*Sqrt[c + d*x]*Sqrt[e + f*x])/(8*b^2*d*f) - ((2*a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f))*Sqrt[c + d*x]*(e + f*
x)^(3/2))/(4*b^2*d*f^2) + (C*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*d*f) - ((16*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2
*(C*d*e + c*C*f + 2*B*d*f) - 2*a*b^2*d*f*(C*(d*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)) - b^3*(C*(d*e - c
*f)^2*(d*e + c*f) - 2*d*f*(B*(d*e - c*f)^2 - 4*A*d*f*(d*e + c*f))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*S
qrt[e + f*x])])/(8*b^4*d^(5/2)*f^(5/2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*Sqrt[b*e - a*f]*ArcTanh[(
Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/b^4

Rule 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (A+B x+C x^2\right )}{a+b x} \, dx &=\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac{\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (\frac{3}{2} b (2 A b d f-a C (d e+c f))-\frac{3}{2} b (2 a C d f+b (C d e+c C f-2 B d f)) x\right )}{a+b x} \, dx}{3 b^2 d f}\\ &=-\frac{(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac{\int \frac{\sqrt{e+f x} \left (\frac{3}{4} b (4 b c f (2 A b d f-a C (d e+c f))+a (d e+3 c f) (2 a C d f+b (C d e+c C f-2 B d f)))+\frac{3}{4} b (4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) x\right )}{(a+b x) \sqrt{c+d x}} \, dx}{6 b^3 d f^2}\\ &=\frac{(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt{c+d x} \sqrt{e+f x}}{8 b^3 d^2 f^2}-\frac{(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac{\int \frac{\frac{3}{8} b \left (16 A b^3 c d^2 e f^2-8 a^3 C d^2 f^2 (d e+c f)+2 a^2 b d f \left (4 B d f (d e+c f)+C \left (d^2 e^2+6 c d e f+c^2 f^2\right )\right )+a b^2 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (4 A d f (d e+c f)+B \left (d^2 e^2+6 c d e f+c^2 f^2\right )\right )\right )\right )-\frac{3}{8} b \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) x}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{6 b^4 d^2 f^2}\\ &=\frac{(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt{c+d x} \sqrt{e+f x}}{8 b^3 d^2 f^2}-\frac{(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac{\left (\left (A b^2-a (b B-a C)\right ) (b c-a d) (b e-a f)\right ) \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{b^4}-\frac{\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{16 b^4 d^2 f^2}\\ &=\frac{(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt{c+d x} \sqrt{e+f x}}{8 b^3 d^2 f^2}-\frac{(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}+\frac{\left (2 \left (A b^2-a (b B-a C)\right ) (b c-a d) (b e-a f)\right ) \operatorname{Subst}\left (\int \frac{1}{-b c+a d-(-b e+a f) x^2} \, dx,x,\frac{\sqrt{c+d x}}{\sqrt{e+f x}}\right )}{b^4}-\frac{\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e-\frac{c f}{d}+\frac{f x^2}{d}}} \, dx,x,\sqrt{c+d x}\right )}{8 b^4 d^3 f^2}\\ &=\frac{(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt{c+d x} \sqrt{e+f x}}{8 b^3 d^2 f^2}-\frac{(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{b c-a d} \sqrt{b e-a f} \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{b^4}-\frac{\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{f x^2}{d}} \, dx,x,\frac{\sqrt{c+d x}}{\sqrt{e+f x}}\right )}{8 b^4 d^3 f^2}\\ &=\frac{(4 b d f (2 A b d f-a C (d e+c f))+(b d e-b c f+4 a d f) (2 a C d f+b (C d e+c C f-2 B d f))) \sqrt{c+d x} \sqrt{e+f x}}{8 b^3 d^2 f^2}-\frac{(2 a C d f+b (C d e+c C f-2 B d f)) \sqrt{c+d x} (e+f x)^{3/2}}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}-\frac{\left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (C d e+c C f+2 B d f)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (B d e+B c f+2 A d f)\right )-b^3 \left (C (d e-c f)^2 (d e+c f)-2 d f \left (B (d e-c f)^2-4 A d f (d e+c f)\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right )}{8 b^4 d^{5/2} f^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{b c-a d} \sqrt{b e-a f} \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{b^4}\\ \end{align*}

Mathematica [B]  time = 6.19865, size = 1944, normalized size = 4.32 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x),x]

[Out]

(2*(A*b^2 - a*b*B + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f)
- (c*d*f)/(d*e - c*f))))^(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c
*f))))) + (Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d
*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 +
(d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2))))/(b^3*Sqrt[d/((d^2*e)/(d*e
- c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*C*(d*e - c*f)*(c + d*x)^(3/2)*Sqrt[e + f
*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(5/2)*((3/(4*(1 + (d*f*(c
+ d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^
2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/2 + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f
))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c
+ d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)
])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((
d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(32*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2
*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2)))/(3*b*d^2*f*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(3/2)
*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*C*e) + b*B*f - a*C*f)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c +
d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f
)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)
)^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c +
d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]
)])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d
^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*
e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*b^2*d*f*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt
[(d*(e + f*x))/(d*e - c*f)]) - ((A*b^2 - a*b*B + a^2*C)*(-(b*c) + a*d)*((2*Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[d/((d^
2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[(d*(e + f*x))/(d
*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e -
c*f)])])/(b*d^(3/2)*Sqrt[e + f*x]) - (2*(-(b*e) + a*f)*ArcTan[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a
*d]*Sqrt[e + f*x])])/(b*Sqrt[-(b*c) + a*d]*Sqrt[b*e - a*f])))/b^3

________________________________________________________________________________________

Maple [B]  time = 0.044, size = 4227, normalized size = 9.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a),x)

[Out]

-1/48*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(-48*A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b
^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f)^(1/2)*a*b^3*c*d^2*f^3-48*A*ln(
(-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-
a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f)^(1/2)*a*b^3*d^3*e*f^2+48*A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*
c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f)^(1/2)*b^
4*c*d^2*e*f^2+24*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*c*d^2*f^3+24*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*
(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*d^3*e*f^2-12*B*ln(1/2*(2
*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/
b^2)^(1/2)*b^4*c*d^2*e*f^2+48*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)
*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f)^(1/2)*a^2*b^2*c*d^2*f^3+48*B*ln((-2*a*d
*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a
*d*e+2*b*c*e)/(b*x+a))*(d*f)^(1/2)*a^2*b^2*d^3*e*f^2-24*B*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f)^
(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b^4*d^2*f^2-24*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*
f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a^2*b^2*c*d^2*f^3-24*C*ln(1/2*(2*
d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b
^2)^(1/2)*a^2*b^2*d^3*e*f^2-6*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(
1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*c^2*d*f^3-6*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x
+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*d^3*e^2*f+3*
C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*
e+b^2*c*e)/b^2)^(1/2)*b^4*c^2*d*e*f^2+3*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*
e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*c*d^2*e^2*f-16*C*x^2*b^4*d^2*f^2*(d*f)^(1/2)
*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)+12*C*((a^2*d*f-a*b*c*f-a*b*d*e+
b^2*c*e)/b^2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*b^3*c*d*f^2+12*C*((a^2*d*f-a*b*c*f-a*b*d*e+b
^2*c*e)/b^2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*b^3*d^2*e*f-4*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2
*c*e)/b^2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*c*d*e*f+24*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*
e)/b^2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*a*b^3*d^2*f^2-4*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*
e)/b^2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b^4*c*d*f^2+48*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*
((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*
(d*f)^(1/2)*a^2*b^2*c*d^2*e*f^2-3*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*
f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*c^3*f^3-48*A*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^
2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*d^2*f^2+6*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(
1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*c^2*f^2+6*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)
*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*d^2*e^2+48*A*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2
)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*d^3*f^3-24*A*ln(1/2*(2
*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/
b^2)^(1/2)*b^4*c*d^2*f^3-24*A*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/
2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*d^3*e*f^2-48*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*
f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f)^(1
/2)*a^3*b*c*d^2*f^3-48*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x
^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f)^(1/2)*a^3*b*d^3*e*f^2+48*B*((a^2*d*f-a*b*c*f-a
*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a*b^3*d^2*f^2-12*B*((a^2*d*f-a*b*c*f-a*
b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*c*d*f^2-12*B*((a^2*d*f-a*b*c*f-a*b*d
*e+b^2*c*e)/b^2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b^4*d^2*e*f-48*C*((a^2*d*f-a*b*c*f-a*b*d*e+
b^2*c*e)/b^2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*a^2*b^2*d^2*f^2-3*C*ln(1/2*(2*d*f*x+2*(d*f*x^2
+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*d^
3*e^3+48*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x
+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f)^(1/2)*a^4*d^3*f^3-48*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^
2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f
)^(1/2)*a*b^3*c*d^2*e*f^2+12*C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1
/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a*b^3*c*d^2*e*f^2-4*C*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^
2)^(1/2)*(d*f)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b^4*d^2*e*f+48*A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2
*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f)
^(1/2)*a^2*b^2*d^3*f^3-48*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2)
)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*a^2*b^2*d^3*f^3+6*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e
)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b^4*c^2*d*f^3+6*B*ln(1
/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*
c*e)/b^2)^(1/2)*b^4*d^3*e^2*f-48*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1
/2)*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*(d*f)^(1/2)*a^3*b*d^3*f^3+48*C*ln(1/2*(2*d
*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^
2)^(1/2)*a^3*b*d^3*f^3)/(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)/b^5/d^2/f^2/(d*f)^(1/2)/((a^2*d*f-a*b*c*f-a*b*d*e+b^2*
c*e)/b^2)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x} \sqrt{e + f x} \left (A + B x + C x^{2}\right )}{a + b x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)*(f*x+e)**(1/2)/(b*x+a),x)

[Out]

Integral(sqrt(c + d*x)*sqrt(e + f*x)*(A + B*x + C*x**2)/(a + b*x), x)

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Giac [B]  time = 3.32303, size = 1461, normalized size = 3.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a),x, algorithm="giac")

[Out]

1/24*sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(d*x + c)*C*abs(d)/(b*d^4) - (7*C*b^9*c
*d^8*f^4*abs(d) + 6*C*a*b^8*d^9*f^4*abs(d) - 6*B*b^9*d^9*f^4*abs(d) - C*b^9*d^9*f^3*abs(d)*e)/(b^10*d^12*f^4))
+ 3*(C*b^9*c^2*d^8*f^4*abs(d) + 2*C*a*b^8*c*d^9*f^4*abs(d) - 2*B*b^9*c*d^9*f^4*abs(d) + 8*C*a^2*b^7*d^10*f^4*
abs(d) - 8*B*a*b^8*d^10*f^4*abs(d) + 8*A*b^9*d^10*f^4*abs(d) - 2*C*a*b^8*d^10*f^3*abs(d)*e + 2*B*b^9*d^10*f^3*
abs(d)*e - C*b^9*d^10*f^2*abs(d)*e^2)/(b^10*d^12*f^4)) + 2*(sqrt(d*f)*C*a^3*b*c*f*abs(d) - sqrt(d*f)*B*a^2*b^2
*c*f*abs(d) + sqrt(d*f)*A*a*b^3*c*f*abs(d) - sqrt(d*f)*C*a^4*d*f*abs(d) + sqrt(d*f)*B*a^3*b*d*f*abs(d) - sqrt(
d*f)*A*a^2*b^2*d*f*abs(d) - sqrt(d*f)*C*a^2*b^2*c*abs(d)*e + sqrt(d*f)*B*a*b^3*c*abs(d)*e - sqrt(d*f)*A*b^4*c*
abs(d)*e + sqrt(d*f)*C*a^3*b*d*abs(d)*e - sqrt(d*f)*B*a^2*b^2*d*abs(d)*e + sqrt(d*f)*A*a*b^3*d*abs(d)*e)*arcta
n(-1/2*(b*c*d*f - 2*a*d^2*f + b*d^2*e - (sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*b)/(
sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 - b^2*c*d*f*e + a*b*d^2*f*e)*d))/(sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 - b^2*c*d*f*e
+ a*b*d^2*f*e)*b^4*d) - 1/16*(sqrt(d*f)*C*b^3*c^3*f^3*abs(d) + 2*sqrt(d*f)*C*a*b^2*c^2*d*f^3*abs(d) - 2*sqrt(d
*f)*B*b^3*c^2*d*f^3*abs(d) + 8*sqrt(d*f)*C*a^2*b*c*d^2*f^3*abs(d) - 8*sqrt(d*f)*B*a*b^2*c*d^2*f^3*abs(d) + 8*s
qrt(d*f)*A*b^3*c*d^2*f^3*abs(d) - 16*sqrt(d*f)*C*a^3*d^3*f^3*abs(d) + 16*sqrt(d*f)*B*a^2*b*d^3*f^3*abs(d) - 16
*sqrt(d*f)*A*a*b^2*d^3*f^3*abs(d) - sqrt(d*f)*C*b^3*c^2*d*f^2*abs(d)*e - 4*sqrt(d*f)*C*a*b^2*c*d^2*f^2*abs(d)*
e + 4*sqrt(d*f)*B*b^3*c*d^2*f^2*abs(d)*e + 8*sqrt(d*f)*C*a^2*b*d^3*f^2*abs(d)*e - 8*sqrt(d*f)*B*a*b^2*d^3*f^2*
abs(d)*e + 8*sqrt(d*f)*A*b^3*d^3*f^2*abs(d)*e - sqrt(d*f)*C*b^3*c*d^2*f*abs(d)*e^2 + 2*sqrt(d*f)*C*a*b^2*d^3*f
*abs(d)*e^2 - 2*sqrt(d*f)*B*b^3*d^3*f*abs(d)*e^2 + sqrt(d*f)*C*b^3*d^3*abs(d)*e^3)*log((sqrt(d*f)*sqrt(d*x + c
) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2)/(b^4*d^4*f^3)